Interferometric iterative technique with bandwidth and numerical-aperture dependency

ABSTRACT

An interferometric intensity equation includes parameters that depend on bandwidth and numerical aperture. An error function based on the difference between actual intensities produced by interferometry and the intensities predicted by the equation is minimized iteratively with respect to the parameters. The scan positions (i.e., the step sizes between frames) that minimized the error function are then used to calculate the phase for each pixel, from which the height can also be calculated in conventional manner. As a result, the phase map generated by the procedure is corrected to a degree of precision significantly better than previously possible.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates in general to the field of scanninginterferometry and, in particular, to a new technique for improving theaccuracy of iterative algorithms for interferometric measurementscarried out on an interference microscope with modulation variations ofthe interference signal due to narrowband wavelengths and/or relativelyhigh numerical apertures in addition to the presence of vibrations.

2. Description of the Prior Art

Many algorithms have been developed in the art for calculating surfacetopography from optical interference data recovered from conventionalscanning techniques. In particular, phase-shifting interferometry (PSI)and related techniques are based on changing the phase differencebetween two coherent interfering beams using a single wavelength λ(ideally) and an optical system with zero numerical-aperture in someknown manner, for example by changing the optical path difference (OPD)either continuously or discretely with time. Several measurements oflight intensity with different OPD values, usually equally spaced, at apixel of a photodetector can be used to determine the phase differencebetween the interfering beams at the point on a test surfacecorresponding to that pixel. Based on such measurements at all pixelswith coordinates (x,y), a phase map Φ(x,y) of the test surface can beobtained, from which very accurate data about the surface profile may becalculated using well known algorithms. (For convenience, the term“pixel” is used hereinafter to refer both to a detector pixel and to thecorresponding region of the sample surface. Also, the term “narrowband”is used exclusively to refer to a light with a spectral bandwidth, asopposed to a theoretically precise, single-wavelength source.)

Several factors, all well understood in the art, affect the quality ofPSI interferometric measurements. For example, a fixed scanning stepsize between acquisition frames is generally assumed for all algorithms,but any factor that causes a change in step size (such as equipmentvibrations, scanner nonlinearities, air turbulence, or wavelengthvariations in the case of wavelength scanning) can affect theperformance of the algorithm and produce a non-uniform profile even whenthe sample surface is perfectly flat. Single-wavelength illumination andzero numerical aperture are also assumed for all algorithms, but notactually used in practice.

PSI corrects fairly well for miscalibrations and slow changes in stepsize, but it is ineffective for random changes in step size. Thus,algorithms based on an iterative approach have been developed toameliorate this problem. See, for example, G. Lai et al., “GeneralizedPhase-Shifting Interferometry,” J. Opt. Soc Am. A, Vol. 8, No. 5, May1991, p. 822; K. Okada et al., “Simultaneous Calculation of PhaseDistribution and Scanning Phase Shift in Phase Shifting Interferometry,”Optics Communications, Vol. 84, Nos. 3-4, July 1991, p. 118; C. Wei etal. “General Phase-Stepping Algorithm with Automatic Calibration ofPhase Step,” Opt. Eng. 38(8), August 1999, pp. 1357-1360; R. Onodera etal., “Phase-Extraction Analysis of Laser-Diode Phase-ShiftingInterferometry that is Insensitive to Changes in Laser Power,” J. Opt.Soc Am. A, Vol. 13, No. 1, January 1996, p. 139; X. Chen et al.,“Phase-Shifting Interferometry with Uncalibrated Phase Shifts,” AppliedOptics, Vol. 39, No. 4, February 2000, p. 585; H. Guo et al.,“Least-Squares Algorithm for Phase-Stepping Interferometry with anUnknown Relative Step,” Applied Optics, Vol. 44, No. 23, August 2005, p.4854; H. Y. Yun et al., “Interframe Intensity Correlation Matrix forSelf-Calibration in Phase-Shifting Interferometry,” Applied Optics, Vol.44, No. 23, August 2005, p. 4860; I. Kong et al., “General Algorithm ofPhase-Shifting Interferometry by Iterative Least-Squares Fitting,”Optical Engineering, Vol. 34, No. 1, January 1995, p. 183. In essence,all of these techniques involve a process whereby the result produced bythe algorithm is refined by iteratively calculating improved step sizes,and correspondingly improved phases, that conform to the actualmodulation data produced by the interferometric measurement. Theseiterative techniques typically find the phase and step size values thatminimize an error function based on the difference between measured andtheoretical intensity, the latter being calculated on the basis ofequations with parameters expressed in function of x,y (sample pixelposition) and z (vertical scan position).

All algorithms assume that the amplitude of modulation remains constantduring the scan, but in fact that is almost never the case. In practice,the light intensity detected as a result of interference of the test andreference beams, which would be perfectly sinusoidal under idealsingle-wavelength and zero-numerical-aperture conditions, exhibits amodulation variation that affects the interferometric result. FIGS. 1Aand 1B illustrate this undesirable condition as might be seen at eachpixel of the interferometer's detector. Thus, the effectiveness of thealgorithm tends to decrease with increased spectral bandwidth of thesource and also with increased numerical aperture of the objective usedfor the measurement.

In practice, a narrowband source (such as a laser) or a filteredbroadband light is used instead of an ideal single-wavelength source tocarry out the interferometric measurement, thereby affecting theamplitude of modulation of the interference signal and the performanceof the algorithm. All systems also utilize a non-zero numericalaperture, which similarly affects the amplitude of modulation of theinterference signal. Therefore, the effectiveness of all prior artalgorithms tends to be undermined by the practical conditions underwhich they are normally implemented. PSI algorithms, which are designedto tolerate fairly well miscalibrated or slow changes in step size, tendto tolerate fairly well also variations of modulation in theinterference signal caused by bandwidth and numerical aperture. However,they are very intolerant of random steps in the optical path of theinterferometer due to mechanical vibration. (Mechanical vibrations areespecially common when interference microscopes are employed inmanufacturing environments.) The iterative techniques, on the otherhand, are tolerant of random steps, but are very intolerant ofvariations in modulation of the interference signal.

In practice, all iterative algorithms used in the art neglect the effectthat numerical apertures (in the order of 0.13 and greater) combinedwith narrowband wavelengths (in the order of 5 nm and greater, ratherthan single) have on the amplitude of the interference signal.Theoretically, an iterative procedure applied to a system with zeronumerical aperture could produce reliable results with illumination ofup to about 5-nm spectral bandwidth, and a system with single wavelengthcould produce reliable results with numerical apertures up to about0.33. However, practical combinations of bandwidth and numericalaperture will produce a significant variation in the modulationamplitude of the interference signal. Thus, it would be very desirableto have an interferometric processing algorithm that accounted for thebandwidth of the light source and the numerical aperture that areactually used in interferometry. The present invention is directed atimproving the prior art by introducing new parameters to reflect theeffects of bandwidth and non-zero numerical aperture in the theoreticalequations used to process the interferometric data.

The following derivation illustrates in detail the general approach usedin the art to formulate the theoretical relationship between phase,scanning position and the intensity measured by the detector at eachpixel as a result of an interferometric measurement. In addition,according to the invention, the approach is expanded to account for theeffects of narrowband light and non-zero numerical aperture. Whilebelieved to be novel, this derivation is presented in the backgroundsection of this disclosure, rather than in the detailed description, inorder to provide a foundation upon which the prior art as well as thepresent invention may be described. Though one skilled in the art willreadily understand that the derivation is not unique, it isrepresentative of the general theoretical equations that can bedeveloped for the relationship between light intensity, phase, scanningposition, numerical aperture and bandwidth.

The following general equation is generally used to represent therelationship between the intensity I of a light source of wavelength λ₀measured at each pixel and the scanning position z_(k) at eachacquisition frame k:

$\begin{matrix}{{{I\left( {x,y,z_{k},\lambda_{0}} \right)} = {B + {M\; {\cos \left( {\frac{4\; \pi}{\lambda_{0}}\left( {h_{obj}^{0} - h_{ref}^{0} - z_{k}} \right)} \right)}}}},} & (1)\end{matrix}$

where I is a function of surface position x,y, scanning position z_(k),and wavelength λ₀; h_(obj) ⁰ and h_(ref) ⁰ are the height positions ofthe object and reference surfaces, respectively, at the beginning of thescan; and B and M are background and modulation amplitude parameters,functions of x, y, z_(k) and λ₀. Note that z_(k) is also a measure ofthe optical path difference between the reference and test beams of theinterferometer. Therefore, while z_(k) is used for conveniencethroughout this disclosure, it is intended also to refer more generallyto OPD in any interferometric system, such as ones where interference isproduced without an actual scan of the object surface. For the sake ofprecision, note also that the term I of Equation 1 expresses irradiance,rather than intensity, but intensity is normally used in the art torefer to both; therefore, intensity will be used throughout thedescription of the invention. Assuming that a fixed, single wavelengthλ₀ is used to perform phase-shifting interferometry, Equation 1 may besimplified by removing the explicit dependence on wavelength, whichyields

$\begin{matrix}\begin{matrix}{{I\left( {x,y,z_{k}} \right)} = {{B\left( {x,y,z_{k}} \right)} +}} \\{{{M\left( {x,y,z_{k}} \right)}{\cos \left( {\frac{4\; \pi}{\lambda_{0}}\left( {h_{obj}^{0} - h_{ref}^{0} - z_{k}} \right)} \right)}}}\end{matrix} & (2)\end{matrix}$

In practice the wavelength of the illumination source is known to spanover some narrow band centered around λ₀, such as illustrated in FIG. 2.Assuming such an asymmetrical illuminating spectrum of bandwidth Wspanning from (λ₀−λ_(M)) to (λ₀+λ_(M)) around the center wavelength λ₀,the sum of the contribution from each wavelength equidistant from λ₀ canbe written (from Equation 2) as follows:

$\begin{matrix}\begin{matrix}{{I\left( {x,y,z_{k},\lambda} \right)} = {B + {M\; {\cos \left( {\frac{4\; \pi}{\lambda_{0} - \lambda}\left( {h_{obj}^{0} - h_{ref}^{0} - z_{k}} \right)} \right)}} +}} \\{{N\; {\cos \left( {\frac{4\; \pi}{\lambda_{0} + \lambda}\left( {h_{obj}^{0} - h_{ref}^{0} - z_{k}} \right)} \right)}}}\end{matrix} & (3)\end{matrix}$

where A is the distance of each wavelength from λ₀ (i.e., referring toFIG. 2, the two wavelengths at λ₀−λ and λ₀+λ), B=B (x,y,z_(k),λ),M=M(x,y,z_(k),λ), N=N(x,y,z_(k),λ), |λ|≦λ_(M), B is background, and Mand N represent signal modulation for the wavelengths λ₀−λ and λ₀+λ oneach side of the center wavelength λ₀.

Equation 3 may be simplified by normalizing its terms by defining

${h \equiv \frac{\lambda}{\lambda_{0}}},{h_{M} \equiv \frac{\lambda_{M}}{\lambda_{0}}},{\phi \equiv {\phi \left( {x,y} \right)} \equiv {\frac{4\; \pi}{\lambda_{0}}\left( {h_{obj}^{0} - h_{ref}^{0}} \right)}},{and}$$t_{k} \equiv {\frac{4\; \pi}{\lambda_{0}}{z_{k}.}}$

For the case where

${0 \leq \frac{\lambda}{\lambda_{0}} \leq \frac{\lambda_{M}}{\lambda_{0}}} = {h_{M} \leq 0.1}$

(such as when a 70 nm bandwidth filter is used centered on 700 nm—notethat in the art filter bandwidth refers to its width around the centerwavelength, i.e., λ₀−λ_(M)), the equation can be further simplified byusing the first order Taylor expansion of the quantities 1/(λ₀−λ) and1/(λ₀+λ), that is

${\frac{1}{\lambda_{0} - \lambda} \cong {\frac{1}{\lambda_{0}}\left( {1 + \frac{\lambda}{\lambda_{0}}} \right)}} = {\frac{1}{\lambda_{0}}\left( {1 + h} \right)}$${\frac{1}{\lambda_{0} + \lambda} \cong {\frac{1}{\lambda_{0}}\left( {1 - \frac{\lambda}{\lambda_{0}}} \right)}} = {\frac{1}{\lambda_{0}}{\left( {1 - h} \right).}}$

Straightforward Substitution Yields the Equation

I(x,y,t _(k),λ₀ ,h)=B+M cos[(φ−t _(k))(1+h)]+N cos[(φ−t_(k))(1−h)],  (4)

which can be further simplified, applying the general trigonometricidentity cos(a±b)=cos(a)cos(b)∓sin(a)sin(b), as follows

I(x,y,t _(k),λ₀ ,h)=B−{(M−N)sin[h(φ−t _(k))]}sin(φ−t_(k))+{(M+N)cos[h(φ−t _(k))]}cos(φ−t _(k)).  (5)

Note that Equation 5 reflects an intensity dependency on x, y and z_(k)(through t_(k)) as well as on wavelength spectrum (through h).

Integration of Equation 5 over h from 0 to h_(M) (i.e., integrating theintensity equation over all wavelengths within the spectrum W of FIG.2), leads to the equation

$\begin{matrix}\begin{matrix}{{I\left( {x,y,t_{k},\lambda_{0}} \right)} = {B + \left\lbrack {h_{M}\left( {M^{eff} - N^{eff}} \right)} \right\rbrack}} \\{{{\frac{{\cos \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} - 1}{h_{M}\left( {\phi - t_{k}} \right)}{\sin \left( {\phi - t_{k}} \right)}} +}} \\{{\left\lbrack {h_{M}\left( {M^{eff} + N^{eff}} \right)} \right\rbrack \frac{\sin \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}{h_{M}\left( {\phi - t_{k}} \right)}{\cos \left( {\phi - t_{k}} \right)}}}\end{matrix} & (6)\end{matrix}$

where B=B(x,y,t_(k)) now denotes a constant of integration with respectto λ, and M^(eff) (x,y,t_(k)) and N^(eff)=N^(eff) (x,y,t_(k)) areeffective mean values of M(x,y,t_(k),λ) and N(x,y,t_(k),λ),respectively, over the entire A domain. Further, for a given λ₀, bysubstituting according to the following definitions,

$\begin{matrix}\begin{matrix}{P \equiv {P\left( {x,y,t_{k},\lambda_{0}} \right)}} \\{{\equiv {\left\lbrack {h_{M}\left( {M^{eff} - N^{eff}} \right)} \right\rbrack \frac{{\cos \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} - 1}{h_{M}\left( {\phi - t_{k}} \right)}}},{and}}\end{matrix} & \left( {7a} \right) \\\begin{matrix}{Q \equiv {Q\left( {x,y,t_{k},\lambda_{0}} \right)}} \\{{\equiv {\left\lbrack {h_{M}\left( {M^{eff} + N^{eff}} \right)} \right\rbrack \frac{\sin \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}{h_{M}\left( {\phi - t_{k}} \right)}}},}\end{matrix} & \left( {7b} \right)\end{matrix}$

it is possible to write Equation 6 in much simpler form as

I(x,y,t _(k))=B+P sin(φ−t _(k))+Q cos(φ−t _(k))  (8)

The two parameters P and Q reflect the general asymmetry in the spectrumof narrowband light. If the spectrum were symmetrical, a singleparameter would suffice.

Finally, applying the same trigonometric identity mentioned above [i.e.,cos(a±b)=cos(a)cos(b)∓sin(a)sin(b), as well assin(a±b)=sin(a)cos(b)∓cos(a)sin(b), Equation 8 can be written to yield

I(x,y,t _(k))=B+[Q cos(φ)+P sin(φ)]cos(t _(k))+[Q sin(φ)−P cos(φ)]sin(t_(k)),  (9)

thereby separating the dependence on φ from the dependence on t_(k). Ofcourse, the coefficients “P” and “Q” will still be dependent on both φand t_(k) and this dependence will not be separable in general. Equation9 is a theoretical expression of intensity measured at a given pixel x,yas a function of initial phase φ and scanning position t_(k), whereinthe narrowband nature of the light source (rather than singlewavelength) is accounted for by the parameters B, P, and Q. Theseparameters remain dependent on x,y, t_(k) and λ₀.

The overall effect of numerical aperture (NA=sin(Θ_(Max))≧0) on themeasured intensity was estimated by A. Dubois et al. in “PhaseMeasurements with Wide-Aperture Interferometers,” Applied Optics, May2000, page 2326. For cases where NA is small (i.e., cosΘ_(Max)≅1), theseauthors determined that the effect of numerical aperture on lightintensity can be accounted for by a coefficient equal tosin(β*t_(k))/(β*t_(k)), where β=(1−cos(Θ_(Max))]/2. Thus, Equation 9 canbe further modified to account for numerical aperture as follows:

$\begin{matrix}\begin{matrix}{{I_{NA}\left( {x,y,t_{k}} \right)} = \left\{ {B + {\left\lbrack {{Q\; {\cos (\phi)}} + {P\; {\sin (\phi)}}} \right\rbrack {\cos \left( t_{k} \right)}} +} \right.} \\{\left. {\left\lbrack {{Q\; {\sin (\phi)}} - {P\; {\cos (\phi)}}} \right\rbrack {\sin \left( t_{k} \right)}} \right\}*} \\{{\frac{\sin \left( {\beta*t_{k}} \right)}{\beta*t_{k}},}}\end{matrix} & (10)\end{matrix}$

wherein the star symbol is used to denote multiplication and I_(NA) isintensity corrected for the effect of an approximately paraxial aperture(the normal condition for interferometric measurements).

Equation 10 may be simplified by defining

$\begin{matrix}{{{B_{NA} \equiv {B\; \frac{\sin \left( {\beta*t_{k}} \right)}{\beta*t_{k}}}} = {B_{NA}\left( {x,y,t_{k},\lambda_{0}} \right)}},} & \left( {11a} \right) \\\begin{matrix}{Q_{NA} \equiv {Q\; \frac{\sin \left( {\beta*t_{k}} \right)}{\beta*t_{k}}}} \\{= {\left\lbrack {h_{M}\left( {M^{eff} + N^{eff}} \right)} \right\rbrack \frac{\sin \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}{h_{M}\left( {\phi - t_{k}} \right)}\frac{\sin \left( {\beta*t_{k}} \right)}{\beta*t_{k}}}} \\{{= {Q_{NA}\left( {x,y,t_{k},\lambda_{0}} \right)}},{and}}\end{matrix} & \left( {11b} \right) \\\begin{matrix}{P_{NA} \equiv {P\; \frac{\sin \left( {\beta*t_{k}} \right)}{\beta*t_{k}}}} \\{= {\left\lbrack {h_{M}\left( {M^{eff} - N^{eff}} \right)} \right\rbrack \frac{{\cos \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} - 1}{h_{M}\left( {\phi - t_{k}} \right)}\frac{\sin \left( {\beta*t_{k}} \right)}{\beta*t_{k}}}} \\{{= {P_{NA}\left( {x,y,t_{k},\lambda_{0}} \right)}},}\end{matrix} & \left( {11c} \right)\end{matrix}$

where the subscript NA is used to indicate the numerical-aperturedependence of the parameter. Substituting from these definitions,Equation 10 becomes

I _(NA)(x,y,t _(k))=B _(NA) +[Q _(NA) cos(φ)+P _(NA) sin(φ)]cos(t_(k))+[Q _(NA) sin(φ)−P _(NA) cos(φ)]sin(t _(k)).  (12)

Note that Equation 12 also expresses theoretical intensity from anarrow-band light centered around λ₀ measured at a given pixel x,y as afunction of phase p and scanning position t_(k), but now both thenarrowband nature of the light source and numerical aperture areaccounted for through the parameters B_(NA), P_(NA) and Q_(NA). For agiven λ₀, these parameters still remain dependent on x,y and t_(k).

The equation generally used in the art to calculate intensity has theform

I(x,y,t _(k))=A(x,y,t _(k))+V(x,y,t _(k))*cos(φ−t _(k)),  (13)

and the functionality of “A” and “V” has been assumed separable, thatis, A(x,y,t_(k))=A₁(x,y)A₂(t_(k)) and V(x,y,t_(k))=V₁(x,y)V₂(t_(k)). Ifno asymmetry around the center wavelength λ₀ is assumed to be present inthe spectrum of the light used to carry out the interferometricmeasurement, the term P_(NA) in Equation 12 derived above is zero(because M^(eff)=N^(eff)) and the equation reduces to:

I _(NA)(x,y,t _(k))=B _(NA) +[Q _(NA) cos(φ)] cos(t _(k))+[Q _(NA)sin(φ)] sin(t _(k)),  (14)

Again, through trigonometric identity, this equation can be written as

I _(NA)(x,y,t _(k))=B _(NA) +Q _(NA) cos(φ−t _(k)),  (15)

which is identical in form to the conventional relation of Equation 13,except that the parameters B_(NA) and Q_(NA) have been derived so as toimplicitly reflect spectral and numerical-aperture functionality. Forthe general case, the dependencies of Q_(NA) on φ and t_(k) will no beseparable. Because of this identity of forms, Equations 12, 14 and 15will be used herein for consistency of notation, instead of Equation 13,first to explain the conventional approach to correcting height forerrors introduced by nonlinearities and environmental perturbations.Then, the same equations will be used to explain the present inventionand its novel features that distinguish it from all techniquesheretofore used in the art.

Thus, referring to Equation 14, the conventional prior-art approachwould involve separating the dependence of the terms over x,y from thedependence over t_(k) [that is, writingB_(NA)(x,y,t_(k))=B₁(x,y)B₂(t_(k)) andQ_(NA)(x,y,t_(k))=Q₁(x,y)Q₂(t_(k))], so that the equation could bewritten as:

I(x,y,t _(k))=B ₁(x,y,)B ₂(t _(k))+Q ₁(x,y)Q ₂(t _(k))cos(φ)cos(t_(k))+Q ₁(x,y)Q ₂(t _(k))sin(φ)sin(t _(k)).  (16)

This relation can be simplified by defining the quantities

C=C(x,y)≡Q ₁(x,y)cos[φ(x,y)]

S=S(x,y)≡Q ₁(x,y)sin[φ(x,y)]

Cos(t _(k))=Cos[t _(k)(t _(k))]≡Q ₂(t _(k))cos(t _(k))

Sin(t _(k))=Sin[t _(k)(t _(k))]≡Q ₂(t _(k))sin(t _(k))

such that Equation 16 can be written as

I(x,y,t _(k))=B ₁(x,y)B ₂(t _(k))+C(x,y)*Cos(t _(k))+S(x,y)*Sin(t_(k))  (17)

In this equation the explicit dependence on p of Equation 16 is removedand retained only implicitly in the substituted parameters C and S.Equation 17 is then used globally (i.e., over x, y and t_(k)) to finditeratively the best sets of parameters [B₁, B₂, C, S, Cos(t_(k)),Sin(t_(k))], and correspondingly the best sets of step sizes and phases,that match the modulation data registered during the interferometricscan.

To that end, an error function T is defined as the difference betweenthe recorded values of intensity and the values predicted by Equation17. Using a conventional least-squares approach, the error function maybe defined as:

$\begin{matrix}\begin{matrix}{{T\left( {x,y,t_{k}} \right)} \equiv {\sum\limits_{x}{\sum\limits_{y}{\sum\limits_{k}\left\lbrack {{I\left( {x,y,t_{k}} \right)} -} \right.}}}} \\{{{{B_{1}\left( {x,y} \right)}{B_{2}\left( t_{k} \right)}} - {{C\left( {x,y} \right)}*{{Cos}\left( t_{k} \right)}} -}} \\{\left. {{S\left( {x,y} \right)}*{{Sin}\left( t_{k} \right)}} \right\rbrack^{2}.}\end{matrix} & (18)\end{matrix}$

Using Equation 18, the set of coordinates [B₁, C, S] and [B₂,Cos(t_(k)), Sin(t_(k))] that minimize the error function T is determinediteratively in conventional manner by the following procedure, which iswell documented in the literature.

Step A. The procedure is started with a guess solution [B₂ ⁰, Cos(t_(k)⁰), Sin(t_(k) ⁰)]. The best set [B₁ ^(i), C^(i), S^(i)] is found thatminimizes the error function T (where the superscript “i” is used todenote the iteration number and i=1 in the first instance). This may bedone in conventional manner by equating partial derivatives of T to zeroand solving for the respective parameters. That is, each of thefollowing identities is solved for its respective unknown variable, B₁^(i), C^(i), and S^(i):

$\begin{matrix}{\frac{\partial T}{\partial B_{1}} = {\frac{\partial T}{\partial C} = {\frac{\partial T}{\partial S} = 0.}}} & (19)\end{matrix}$

Step B. Using the newly determined values for [B₁ ^(i), C^(i), S^(i)],the error function T is updated. Then, a set [B₂ ^(i), Cos(t_(k) ^(i)),Sin(t_(k) ^(i))] that minimizes the error function is similarlydetermined. This may be done again by solving the partial derivativeequations

$\begin{matrix}{\frac{\partial T}{\partial B_{2}} = {\frac{\partial T}{\partial{{Cos}\left( t_{k} \right)}} = {\frac{\partial T}{\partial{{Sin}\left( t_{k} \right)}} = 0.}}} & (20)\end{matrix}$

Step C. Next, the step size t_(k) ^(i) (i.e., the frame separationduring the scan) is estimated from the usual relation

$\begin{matrix}{{t_{k}^{i} = {\arctan \left( \frac{{{Sin}\left( t_{k} \right)}^{i}}{{{Cos}\left( t_{k} \right)}^{i}} \right)}},} & (21)\end{matrix}$

and t_(k) ^(i) is compared to the value obtained in the previous step.Initially, it is compared to the scanner's design step size, which willalso be the step size for the interferometric algorithm in use(typically π/2). If the difference is less than a predetermined limit,the iteration process is stopped. Otherwise, the process is repeatedfrom Step A, wherein the values of [B₂ ^(i+1), Cos(t_(k) ^(i+1)),Sin(t_(k) ^(i+1))] are updated using the previous solution [B₂ ^(i),Cos(t_(k) ^(i), Sin(t_(k) ^(i))]

Step D. When the iterative process is stopped as described above, thephase (and correspondingly the height) is calculated for each pixel x,yin conventional manner using the equation

$\begin{matrix}{{\phi = {{\arctan \left\lbrack \frac{\sin \; \phi}{\cos \; \phi} \right\rbrack} = {\arctan \left( \frac{S^{i}}{C^{i}} \right)}}},} & (22)\end{matrix}$

where i corresponds to the last step of iteration.

The conventional technique described above produces good results fornarrowband light that approaches single wavelength illumination and forobjectives without low numerical aperture. As the spectral band of thelight source and/or the numerical aperture increase, the performance ofthe techniques currently in use deteriorates rapidly and the results areno longer reliable. For example, spectral bands in the order of 5 nm orgreater produce a significant amount of modulation variation in theintensity data, which in turn causes erroneous height calculations.(Note that modulation variation is also commonly referred to asamplitude modulated sinusoidal signal in optics and electricalengineering.) Similar results are produced when numerical aperturesgreater than about 0.33 are used. Therefore, any procedure that allowedcorrection when narrowband light and/or significant numerical apertureare used would constitute a very useful advance in the art.

BRIEF SUMMARY OF THE INVENTION

The heart of the invention is in the idea of employing an equation,preferably based on theoretical considerations, that expresses theintensity measured at each pixel during an interferometric scan as afunction of scan position, initial phase map, and a set of parametersthat depend on the bandwidth and the numerical aperture of the systemused to acquire the interferometric data. Such an equation is used toestablish a global error function that can thus be minimized for alldata with respect to the best parameters that produce a calculatedmodulation that most closely approximates the interferometric data. Oncethese parameters are found, the scan positions (i.e., the step sizesbetween frames) that produced such best approximation are used tocalculate the phase for each pixel, from which the height can also becalculated in conventional manner. The measurement corrections enabledby this process can substantially reduce the impact of mechanicalvibrations, scanner nonlinearities, air turbulence, or wavelengthvariations that can otherwise afflict the measurements of interferencemicroscopes.

The preferred error function is a least-squares representation of thedifference between the actual intensities produced by theinterferometric measurement and the intensities predicted by thetheoretical equation. According to one aspect to the invention, theerror function minimization is carried out iteratively by minimizingsequentially with respect to two subsets of parameters, wherein at eachiteration the values produced by the previous iteration for the firstsubset are used to calculate the second subset, and then the new valuesfor the second subset are used to update the first subset. The iterativeprocess is continued until the error function is minimized to within apredetermined value deemed acceptable.

The iteration process may be simplified by introducing a transformwhereby two of the parameters in the equation expressing intensityaccording to the invention are normalized. As a result, the phase mapgenerated by the procedure of the invention is corrected to a degree ofprecision significantly better than previously possible.

Various other advantages of the invention will become clear from itsdescription in the specification that follows and from the novelfeatures particularly pointed out in the appended claims. Therefore, tothe accomplishment of the objectives described above, this inventionconsists of the features hereinafter illustrated in the drawings, fullydescribed in the detailed description of the preferred embodiment, andparticularly pointed out in the claims. However, such drawings anddescription disclose but a few of the various ways in which theinvention may be practiced.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates the plot of intensity versus scanning heightregistered at a pixel of an interferometer's detector when asingle-wavelength light source is used to illuminate the sample withzero numerical aperture and no environmental perturbations (such asvariable scanning steps, vibrations, and air turbulence).

FIG. 1B shows the modulation effect produced by a narrow-band lightsource and non-zero numerical aperture on the intensity plot of FIG. 1.

FIG. 2 is an illustration of the spectral distribution of a typicalnarrow-band light source used in phase-shifting interferometry.

FIG. 3 is block diagram of the broad steps involved in carrying out theinvention.

FIG. 4A is the profile of a flat surface obtained with 20-nm bandwidthillumination, a 0.13 NA, and a conventional eight-frame PSI algorithm.

FIG. 4B is the profile of the flat surface of FIG. 4A obtained with aconventional iterative algorithm.

FIG. 4C is the profile of the flat surface of FIG. 4A obtained,according to the invention, with an algorithm including correction formodulation variations produced by numerical aperture and narrowbandwavelength effects, in addition to scan nonlinearities and environmentalfactors.

DETAILED DESCRIPTION OF THE INVENTION

One aspect of the invention lies in the addition of bandwidth andnumerical-aperture dependence in the parameters of the equations used toexpress the theoretical intensity produced by interference. According toanother aspect, the invention introduces an additional and new step inthe numerical procedure followed to solve for the parameters thatproduce near identity between the theoretical and the actual values ofintensity recorded during the interferometric scan. As a result, thenumerical procedure of the invention corrects interferometric data forscanner nonlinearities and environmental perturbations, as well as forerrors caused by narrowband light and significant numerical aperture.

As used herein, the term “global,” when used with reference to a set ofvariables, is intended to refer to all data points acquired over thespace spanned by the set of variables. For instance, minimization of aglobal error function wherein x, y and z are variables is intended torefer to a minimization with respect to given parameter over all dataacquired in x, y and z space.

Referring back to Equation 12, it may be simplified by defining andsubstituting the following quantities:

U(x,y,t _(k))=Q _(NA) cos(φ)+P _(NA) sin(φ) and  (23a)

V(x,y,t _(k))=Q _(NA) sin(φ)−P _(NA) cos(φ),  (23b)

where U and V depend upon x, y and t_(k). This substitution removes theexplicit dependence of U and V on φ, which is instead only implicitlypresent in the parameters. In order to further separate the dependenceof U and V from x,y from their dependence from t_(k), these parametersare then expressed in the following manner:

U(x,y,t _(k))=U ₁(x,y)*U ₂(t _(k))*f _(c)(x,y,t _(k))  (24a)

V(x,y,t _(k))=V ₁(x,y)*V ₂(t _(k))*f _(x)(x,y,t _(k))  (24b)

where the new factors f_(c)(x,y,t_(k)) and f_(x)(x,y,t_(k)) representcoefficients correlating the inseparable dependency of U and V from x,yand t_(k), but U₁, U₂, V₁ and V₂ reflect separate dependencies. Thus,Equation 12 becomes

I _(NA)(x,y,t _(k))=B _(NA) +U ₁(x,y)U ₂(t _(k))f _(c)(x,y,t _(k))cos(t_(k))+V ₁(x,y)V ₂(t _(k))f _(x)(x,y,t _(k))sin(t _(k)).  (25)

The following additional identities are then defined to simplifyEquation 25:

B _(NA)(x,y,t _(k))=B ₁(x,y)B ₂(t _(k)),

C(x,y,t _(k))≡U ₁(x,y)*f _(c)(x,y,t _(k))

S(x,y,t _(k))≡V ₁(x,y)*f _(x)(x,y,t _(k)),

Cos[t _(k)(t _(k))]≡U ₂(t _(k))cos(t _(k)), and

Sin[t _(k)(t _(k))]≡V ₂(t _(k))sin(t _(k))  (26)

Thus, Equation 25 may be written as:

I(x,y,t _(k))=B ₁(x,y)B ₂(t _(k))+C(x,y,t _(k))*Cos(t _(k))+S(x,y,t_(k))*Sin(t _(k))  (27)

Note that the difference between Equation 27 and Equation 17 above (theprior-art equation with the same form) lies only in the implicitdependence of the “C” and “S” parameters on the “t_(k)” variable. Thatis, the two parameters, C=C(x,y,t_(k)) [instead of C=C(x,y)] andS=S(x,y,t_(k)) [instead of S=S(x,y)], are the items that distinguish theequation of the invention from the prior art. As mentioned before, thisdependency is introduced to account for the wavelength bandwidth and thenumerical aperture.

According to the invention, Equation 27 is used globally (i.e., over x,yand t_(k)) to find iteratively the best sets of parameters, steps sizesand phases that match the modulation data registered during theinterferometric scan. To that end, an error function T′ is defined, asin the prior art, on the basis of the difference between the recordedvalues of intensity and the values predicted by Equation 27. Using againa least-squares approach, the error function T′ may be written as:

$\begin{matrix}\begin{matrix}{{T^{\prime}\left( {x,y,t_{k}} \right)} \equiv {\sum\limits_{x}{\sum\limits_{y}{\sum\limits_{k}\left\lbrack {{I\left( {x,y,t_{k}} \right)} -} \right.}}}} \\{{{{B_{1}\left( {x,y} \right)}{B_{2}\left( t_{k} \right)}} -}} \\{{{{C\left( {x,y,t_{k}} \right)}*{{Cos}\left( t_{k} \right)}} -}} \\{\left. {{S\left( {x,y,t_{k}} \right)}*{{Sin}\left( t_{k} \right)}} \right\rbrack^{2}.}\end{matrix} & (28)\end{matrix}$

The set of coordinates [B¹, C, S] and [B₂, Cos(t_(k)), Sin(t_(k))] thatminimize the error function T′ is determined iteratively, as in theprior-art procedure, but with the novel steps outlined below. Adescription of the iterative process follows.

Step A. Using a guess solution for [B₂ ⁰, Cos(t_(k) ⁰), Sin(t_(k) ⁰)],the best set [B₁ ^(i), C^(i), S^(i)] that minimizes the error functionT′ is found (wherein “i” is used again to denote iteration number). Thisis also preferably done in conventional manner by solving the partialderivative equations

$\begin{matrix}{\frac{\partial T^{\prime}}{\partial B_{1}} = {\frac{\partial T^{\prime}}{\partial C} = {\frac{\partial T^{\prime}}{\partial S} = 0.}}} & (29)\end{matrix}$

Step B. At this point of the iteration, the goal is to determine [B₂^(i), Cos(t_(k) ^(i)), Sin(t_(k) ^(i))] based on [B₁ ^(i), C^(i), andS^(i)] and [B₂ ^(i−1), Cos(t_(k) ^(i−1)), Sin(t_(k) ^(i−1))], which areknown. To that end, it is desirable to decouple the x,y dependence of Cand S from their t_(k) dependence (i.e., their planar position from thescanning height). This can be achieved by expressing Equation 27 in theform

I(x,y,t _(k))=B ₁ ^(i)(x,y)B ₂ ^(i)(t _(k))+C ₁ ^(i)(x,y)*Cos(t _(k)^(i))+S₁ ^(i)(x,y)*Sin(t _(k) ^(i)),  (27′)

where C₁ ^(i) and S₁ ^(i) are new values that for each pixel depend onlyon phase, φ, and do not depend upon scanning position, t_(k). To achievethis objective, according to another aspect of the present invention,the set of parameters [C^(i), S^(i)] produced at each iteration isassumed to be a linear combination of the actual cos(φ) and sin(φ)values multiplied by some arbitrary coefficients. Thus, each set ofparameters [C^(i), S^(i)] produced by the step is expressed in terms ofa new set [C₁ ^(i), S₁ ^(i)], as follows:

C ^(i) =aC ₁ ^(i) +bS ₁ ^(i)  (30a)

S ^(i) =aS ₁ ^(i−bC) ₁ ^(i),  (30b)

wherein C₁ ^(i) ad S₁ ^(i) are terms that reflect the combineddependence of C and S on both sine and cosine of phase [i.e.,C^(i)˜a*sin(φ)+b*cos(φ) and S^(i)˜a*sin(φ)−b*cos(φ)]. Note the U and Vdependence of C and S (from Equations 26 and 24 above), the Q_(NA) andP_(NA) dependence of U and V (from Equations 23), and the sin(φ) andcos(φ) dependence of Q_(NA) and P_(NA) (developed for the theoreticalintensity generated by a narrow-band source of wavelength centeredaround λ₀—see Equations 11). Therefore, the assumption that [C^(i),S^(i)] may be expressed as a linear combination of the actual cos(φ) andsin(φ) values multiplied by some arbitrary coefficients is well foundedin the derivation of the equations used to practice the invention. Thus,while φ=arctan[S/C] would produce an approximate value of phase, a moreprecise value is expected to result from solving the implicit equation

φ=arctan[S ₁ /C ₁]  (31)

Inverting Equations 30 yields the following expressions for C₁ ^(i) andS₁ ^(i) as functions of a and b:

$\begin{matrix}\left\{ \begin{matrix}{C_{1}^{i} = \frac{{aC}^{i} - {bS}^{i}}{a^{2} + b^{2}}} \\{S_{1}^{i} = \frac{{aS}^{i} + {bC}^{i}}{a^{2} + b^{2}}}\end{matrix} \right. & (32)\end{matrix}$

Replacing these relations for C₁ ^(i) and S₁ ^(i) in Equation 27′ yields

$\begin{matrix}\begin{matrix}{{I\left( {x,y,t_{k}} \right)} = {{{B_{1}^{i}\left( {x,y} \right)}{B_{2}^{i - 1}\left( t_{k} \right)}} +}} \\{{{\frac{{aC}^{i} - {bS}^{i}}{a^{2} + b^{2}}*{{Cos}\left( t_{k}^{i - 1} \right)}} +}} \\{{{\frac{{aS}^{i} + {bC}^{i}}{a^{2} + b^{2}}*{{Sin}\left( t_{k}^{i - 1} \right)}},}}\end{matrix} & (33)\end{matrix}$

where a and b are the only unknown values. All the other parameters areknown. In order to simplify the iterative solution for a and b, notingthat C₁ and S₁ are assumed proportional to cosine and sine functions,respectively, and that sin²+cos²=1, the following condition is imposed:

(C ₁ ^(i))²+(S ₁ ^(i))² =A ²=const.  (34)

Squaring both sides of Equations 30 and summing them yields

[S ^(i)]² +[C ^(i)]² =a ²([S ₁ ^(i)]² +[C ₁ ^(i)]²)+b ²([S ₁ ^(i)]²+[C₁^(i)]²)  (35)

Combining Equations 34 and 35 produces

[S ^(i)]² +[C ^(i)]²=(a ² +b ²)A ²,  (36)

that is,

$\begin{matrix}{{a^{2} + b^{2}} = {\frac{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}{A^{2}}.}} & (37)\end{matrix}$

Replacing Equation 37 in Equation 33 produces the following relation:

$\begin{matrix}\begin{matrix}{{I\left( {x,y,t_{k}} \right)} = {{{B_{1}^{i}\left( {x,y} \right)}{B_{2}^{i - 1}\left( t_{k} \right)}} +}} \\{{{\frac{A^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {{aC}^{i} - {bS}^{i}} \right)*{{Cos}\left( t_{k}^{i - 1} \right)}} +}} \\{{\frac{A^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {{aS}^{i} + {bC}^{i}} \right)*{{Sin}\left( t_{k}^{i - 1} \right)}}} \\{= {{{B_{1}^{i}\left( {x,y} \right)}{B_{2}^{i - 1}\left( t_{k} \right)}} +}} \\{{{\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {C^{i} - {\frac{b}{a}S^{i}}} \right)*{{Cos}\left( t_{k}^{i - 1} \right)}} +}} \\{{\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {S^{i} + {\frac{b}{a}C^{i}}} \right)*{{Sin}\left( t_{k}^{{ii} - 1} \right)}}}\end{matrix} & (38)\end{matrix}$

By defining the quantity t≡b/a and substituting it in Equation 38, oneobtains

$\begin{matrix}\begin{matrix}{{I\left( {x,y,t_{k}} \right)} = {{{B_{1}^{i}\left( {x,y} \right)}{B_{2}^{i - 1}\left( t_{k} \right)}} +}} \\{{{\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {C^{i} - {tS}^{i}} \right)*{{Cos}\left( t_{k}^{i - 1} \right)}} +}} \\{{{\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {S^{i} + {tC}^{i}} \right)*{{Sin}\left( t_{k}^{i - 1} \right)}},}}\end{matrix} & (39)\end{matrix}$

with the two new variables “aA²” and “t.” Forming a new error functionT″ on the basis of Equation 39 yields

$\begin{matrix}\begin{matrix}{{T^{''}\left( {x,y,t_{k}} \right)} \equiv {\sum\limits_{x}{\sum\limits_{y}{\sum\limits_{k}\left\lbrack {{I\left( {x,y,t_{k}} \right)} - {{B_{1}^{i}\left( {x,y} \right)}{B_{2}^{i - 1}\left( t_{k} \right)}} -} \right.}}}} \\{{\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {C^{i} - {tS}^{i}} \right)*{{{Cos}\left( t_{k}^{i - 1} \right)}--}}} \\{\left. {\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {S^{i} + {tC}^{i}} \right)*{{Sin}\left( t_{k}^{i - 1} \right)}} \right\rbrack^{2}.}\end{matrix} & (40)\end{matrix}$

Equation 40 can then be used explicitly to solve for “aA²” and “t”, byimposing the usual conditions:

$\begin{matrix}{{\frac{\partial T^{''}}{\partial\left( {aA}^{2} \right)} = {\frac{\partial T^{''}}{\partial t} = 0}},} & (41)\end{matrix}$

which provide two equations with two unknowns. Once the quantities aA²and t are so determined for the i^(th) iteration, they are plugged backinto Equation 40 to continue the iterative process.

Step C. Using the values so determined for (aA²)^(i) and t^(i), theerror function T″ is updated. Then, a new set [B₂ ^(i), Cos(t_(k))^(i),Sin(t_(k))^(i)] that minimizes the updated error function is determined,again by solving the derivative equations:

$\begin{matrix}{\frac{\partial T^{''}}{\partial B_{2}} = {\frac{\partial T^{''}}{{\partial{Cos}}\; t_{k}} = {\frac{\partial T^{''}}{{\partial S}\; {int}_{k}} = 0.}}} & (42)\end{matrix}$

Step D. Next, the step size t_(k) ^(i) (the frame separation during thescan) is estimated from the relation

$\begin{matrix}{{t_{k}^{i\;} = {\arctan \left( \frac{{{Sin}\left( t_{k} \right)}^{i}}{{{Cos}\left( t_{k} \right)}^{i}} \right)}},} & (43)\end{matrix}$

and t_(k) ^(i) is compared to the value obtained in the previous step.If the difference is less than a predetermined limit, the iterationprocess is stopped. Otherwise, the process is repeated from step A.

Step E. When the iterative process is concluded, the phase (andcorrespondingly the height) is calculated for each pixel x,y using theequation

$\begin{matrix}{{\phi = {\arctan \left( \frac{S^{i}}{C^{i}} \right)}},} & (44)\end{matrix}$

where i corresponds to the last step of iteration.

One skilled in the art will recognize that the novelty of the inventionlies in step B, which is an additional step within the standarditerative algorithm. In essence, it involves the addition of parametersthat account for wavelength and numerical-aperture dependencies in theequations used to express the theoretical intensity measured during aninterferometric measurement. These parameters yield a “phase correction”for bandwidth and numerical aperture that was not previously achieved inthe art. In addition, the step involves a phase-dependent parametersubstitution that permits the rational decoupling of the dependence of Cand S from the planar coordinates (x,y) and the scanning heightcoordinate (t_(k)). These new phase-dependent parameters in theequations simplify the numerical process and make it possible to refinethe phase correction to a level significantly better than previouslypossible using conventional iterative techniques. FIG. 3 is a blockdiagram of the general procedure of the invention.

FIGS. 4A, 4B and 4C illustrate the corrective power of the algorithm ofthe invention compared to conventional algorithms. FIG. 4A shows asample profile produced by a typical eight-frame PSI algorithm using a20-nm bandwidth illumination with 0.13 numerical aperture in a noisyenvironment. FIG. 5B shows a profile of the same sample under the sameconditions produced with a prior-art iterative algorithm. (The algorithmdisclosed by Okada et al., supra, was used as an exemplary conventionaliterative algorithm.) FIG. 4C is a profile of the same surface obtainedagain under the same conditions, but using the iterative algorithm ofthe invention that also includes correction for modulation variationscaused by relatively large numerical aperture and by narrowbandwavelength. All figures reflect data taken with the same optical systemand under the same sampling and environmental conditions. The correctionalgorithm offers a substantial reduction in noise by accounting for theeffects of bandwidth and numerical aperture as well as vibrations andother environmental factors.

According to another aspect of the invention, the process may besimplified by normalizing C₁ ^(i) and S₁ ^(i) as follows:

$\begin{matrix}{{C_{1}^{i} = \frac{C^{i}}{\left\lbrack C^{i} \right\rbrack^{2} + \left\lbrack S^{i} \right\rbrack^{2}}},} & \left( {45a} \right) \\{{S_{1}^{i} = \frac{S^{i}}{\left\lbrack C^{i} \right\rbrack^{2} + \left\lbrack S^{i} \right\rbrack^{2}}},{and}} & \left( {45b} \right)\end{matrix}$

and substituting these relations in the error function T″ (Equation 40).In so doing, the error function T″ becomes expressed in terms of C₁ ^(i)and S₁ ^(i) and can be used to perform all relevant steps of theiterative process (that is, C₁ ^(i) and S₁ ^(i) are calculated at eachiteration using T″ instead of C¹ and S^(i) using T′).

Note that this normalization flows naturally from the followingobservations. From the relation of the coefficients a and b definedabove for Equation 39 and their relation to sine and cosine, which issimilar to the relation of the parameters Q and P (see Equations 7above), the following condition can be assumed:

$\begin{matrix}{{\frac{b}{a} \approx \frac{P}{Q} \cong {\frac{M^{eff} - N^{eff}}{M^{eff} + N^{eff}}\frac{{\cos \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} - 1}{\sin \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}} \equiv t},} & (46)\end{matrix}$

where t is constant and in practice is a very small number on account ofthe good symmetry of the light used in interferometry (i.e.,M^(eff)−N^(eff)=0). Furthermore, solving the system of Equations 30a and30b produces the following identities:

$\begin{matrix}{\begin{matrix}{S_{1}^{i} = \frac{{aS}^{i} + {bC}^{i}}{a^{2} + b^{2}}} \\{= {\frac{A^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {{aS}^{i} + {bC}^{i}} \right)}} \\{{= {\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {S^{i} + {\frac{b}{a}C^{i}}} \right)}},{and}}\end{matrix}\begin{matrix}{C_{1}^{i} = \frac{{aC}^{i} - {bS}^{i}}{a^{2} + b^{2}}} \\{= {\frac{A^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}\left( {{aC}^{i} - {bS}^{i}} \right)}} \\{= {\frac{{aA}^{2}}{\left\lbrack S^{i} \right\rbrack^{2} + \left\lbrack C^{i} \right\rbrack^{2}}{\left( {C^{i} - {\frac{b}{a}S^{i}}} \right).}}}\end{matrix}} & (47)\end{matrix}$

If the ratio b/a is very small, as mentioned, and aA² is a fixed number(being by definition the product of two constants), the set of Equations47 can be written as

$\begin{matrix}{C_{1}^{i} \approx \frac{C^{i}}{\left\lbrack C^{i} \right\rbrack^{2} + \left\lbrack S^{i} \right\rbrack^{2}}} & \left( {48a} \right) \\{{S_{1}^{i} \approx \frac{S^{i}}{\left\lbrack C^{i} \right\rbrack^{2} + \left\lbrack S^{i} \right\rbrack^{2}}},} & \left( {48b} \right)\end{matrix}$

which explains the appropriateness of the normalization given above.

While the procedure of the invention given above is general, it ispreferably carried out under conditions most suitable to conform to theassumptions underlying it derivation. In particular, the following datacollection conditions are recommended:

-   -   1. The data are taken symmetrically around best focus (i.e., the        interferometer's objective is at focus in the center of the        image at the middle frame of the vertical scan).    -   2. A narrow bandpass filter (+/−20 nm) is used, such that

${h_{M} \equiv \frac{\lambda_{M}}{\lambda_{0}}} = {\frac{20}{700} = {0.03.}}$

-   -   3. The PSI measurement is done with 3-12 frames separated by

$\frac{\pi}{2}{\left( {{t_{k}} \leq {1.5\pi}} \right).}$

-   -   4. The number of fringes in the field of view is about 5        (|φ|≦5π).        Under these working parameters, the following condition can be        expected to be true

|h _(M)(φ−t _(k))|≦0.03*(5+1.5)*π≅0.2*π  (49)

Referring back to Equation 6, a substitution of the following Taylorexpansions,

$\begin{matrix}{{\sin \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} \cong {{h_{M}\left( {\phi - t_{k}} \right)} - \frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack^{3}}{6}}} & (50) \\\text{and} & \; \\{{{\cos \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} \cong {1 - \frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack^{2}}{2}}},} & \;\end{matrix}$

yields the following equation for the recorded interferometric signal:

$\begin{matrix}{{I\left( {x,y,t_{k},\lambda_{0}} \right)} = {B + {\left( {N^{eff} - M^{eff}} \right)\frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}{2}{\sin \left( {\phi - t_{k}} \right)}} + {{\left( {M^{ef} + N^{ef}} \right)\left\lbrack {1 - \frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack^{2}}{6}} \right\rbrack}{{\cos \left( {\phi - t_{k}} \right)}.}}}} & (51)\end{matrix}$

Defining now new quantities “P” and “Q” such that:

$\begin{matrix}{{P \equiv {\left( {N^{eff} - M^{eff}} \right)\frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}{2}}},} & (52) \\\text{and} & \; \\{{Q \equiv {\left( {M^{eff} + N^{eff}} \right)\left\lbrack {1 - \frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack^{2}}{6}} \right\rbrack}},} & \;\end{matrix}$

the signal intensity of Equation 51 becomes:

I(x,y,t _(k),λ₀)=B+P*sin(φ−t _(k))+Q*cos(φ−t _(k)).  (53)

Expanding the “sin” and “cos” terms in Equation 53 produces thefollowing:

$\begin{matrix}{{{I\left( {x,y,t_{k},\lambda_{0}} \right)} = {B + {{Q\left\lbrack {{\cos (\phi)} + {\frac{P}{Q}{\sin (\phi)}}} \right\rbrack}{\cos \left( t_{k} \right)}} + {{Q\left\lbrack {{\sin (\phi)} - {\frac{P}{Q}{\cos (\phi)}}} \right\rbrack}{\sin \left( t_{k} \right)}}}},} & (54)\end{matrix}$

where

B=B(x,y,t _(k),λ₀),

Q=Q(x,y,t _(k),λ₀),

P=P(x,y,t _(k),λ₀),

φ=φ(x,y).

Defining

$\begin{matrix}\begin{matrix}{R \equiv \frac{P}{Q}} \\{= {\frac{M^{eff} - N^{eff}}{M^{eff} + N^{eff}}*\frac{{\cos \left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack} - 1}{\sin \left\lbrack {h_{M}\left( {\eta - t_{k}} \right)} \right\rbrack}}} \\{{\cong {\frac{M^{eff} - N^{eff}}{M^{eff} + N^{eff}}*\frac{\frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack}{2}}{1 - \frac{\left\lbrack {h_{M}\left( {\phi - t_{k}} \right)} \right\rbrack^{2}}{6}}}},}\end{matrix} & (55)\end{matrix}$

for the case when |h_(M)(φ−t_(k))|≦0.2*π, as described above withreference to the preferred conditions to practice the invention, itfollows that

$\begin{matrix}{{{\frac{M^{eff} - N^{eff}}{M^{eff} + N^{eff}}}*\frac{0.1*\pi}{1 - \frac{0.04*\pi^{2}}{6}}} \cong {0.34*{{\frac{M^{eff} - N^{eff}}{M^{eff} + N^{eff}}}.}}} & (56)\end{matrix}$

For the case of and asymmetry

${{{\frac{M^{eff} - N^{eff}}{M^{eff} + N^{eff}}} \cong {5\%}} = 0.05},$

the following hold true:

0<|R|≦1.7 %,  (57)

which justifies the assumption made above in deriving Equation 39 thatb/a is small.

Thus, a novel iterative procedure has been described that advantageouslyallows correction for modulation variations produced by bandwidth andnumerical-aperture conditions normally encountered in practice. Themethod has been particularly effective for the measurement of opticallysmooth, flat surfaces with steps. For example, magnetic heads used indata storage, electronic substrates in semiconductor components, plasticelectronics components, solar energy conversion devices, micro-mirrorarrays and micro-electro-mechanical systems (MEMS) devices are allcharacterized by such smooth, flat surfaces with steps. As FIGS. 4A-4Cillustrate, the invention enables a measurement with significantlyimproved precision over prior-art iterative algorithms.

It is clear from the derivation of the equation and procedure of theinvention that partial benefits may be achieved by utilizing an equationthat includes a parameter that depends only on bandwidth or only onnumerical aperture. That is, the iterative procedure may be carried outwith respect to a single novel parameter and yet produce improvedresults over the prior art. Therefore, such more limited application ofthe concept described herein is intended to be part of the inventioncovered herein.

While the invention has been shown and described herein in what isbelieved to be the most practical and preferred embodiments, it isrecognized that departures can be made therefrom within the scope of theinvention. For example, the invention has been described with referenceto interferometric microscope applications, but it is clearly applicableas well to other methods that use fringes for analysis (that is,analysis most commonly based on PSI methods). These include, withoutlimitation, digital holography methods that use phase-shifted holograms,wavelength scanning and spectrally resolved white light interferometricsystems, speckle systems, fringe projection methods (in which fringesmay be created in many different ways, such as by projecting fringesfrom an interferometer, a ronchi grating, a digital signal projector, oran OLED device). In essence, the invention can be used advantageouslywhenever PSI methods are used for phase calculations if vibrations andmodulation variations are present. Therefore, the invention is not to belimited to the details disclosed herein but is to be accorded the fullscope of the claims so as to embrace any and all equivalent processesand products.

1. A method of reducing errors in measurements performed with aninterference microscope, the method comprising the following steps:employing an equation espressing intensity as a function of planarcoordinates, optical path difference, phase, and a set of equationparameters reflecting an implicit dependence on illumination spectralbandwidth and numerical aperture, said equation parameters having anexplicit dependence on planar coordinates, optical path difference andphase; replacing said equation parameters with new phase-dependentparameters such that said explicit dependence of the equation parameterson planar coordinates is decoupled from the dependence on optical pathdifference; fitting said equation to global interference data acquiredby varying said optical path difference, thereby producing an optimalset of said equation parameters, said fitting step being an iterativeprocess wherein said equation parameters are estimated to apredetermined iterative threshold of convergence; and calculating amodified phase value using said optimal set of equation paramaters. 2.The me hod of claim 1, further including the step of calculating aheight map of a sample based on said modified phase.
 3. The method ofclaim 1, wherein the calculating stop attenuates an impact of noise inthe measurements due to mechanical vibrations present at theinterference microscope.
 4. The method of claim 2, wherein thecalculating step attenuates an impact of noise in the height map due tomechanical vibrations present at the interference microscope.
 5. Themethod of claim 1, wherein said interference data acquired by varyingthe optical path difference are taken symmetrically around a best focusposition.
 6. The method of claim 1, wherein said interference dataacquired by varying the optical path difference are produce using alight source including a bandpass filter of a proximately 20-nmbandwidth.
 7. The method of claim 1 wherein said interference data areacquired over three to twelve frames separated by a nominal phase stepof π/2.
 8. The me hod of claim 1, wherein said interference microscopeis operated so as to produce about five fringes within a field of viewof an objective of the microscope
 9. The method of claim 1, wherein saidstep of varying the optical path difference is carried out with avertical scan.
 10. The method of claim 1, wherein the method is used tomeasure a magnetic read-write head for data storage.
 11. The method ofclaim 1, wherein the method is used to measure an electronic substratein a semiconductor component.
 12. The method of claim 1, wherein themethod is used to measure a plastic electronics component.
 13. Themethod of claim 1, wherein the method is used to measure a solar energyconversion device.
 14. The method of claim 1, wherein the method is usedto measure a micro-mirror array.
 15. The method of claim 1, wherein themethod is used to measure a MEMS device.
 16. An interferometriciterative procedure comprising: selecting an equation expressingintensity as a function of planar coordinates, optical path difference,phase, and a set of bandwidth-dependent and numerical-aperture-dependentequation parameters, said equation parameters having an explicitdependence on planar coordinates, optical path difference and phase;replacing said equation parameters with new phase-dependent parameterssuch that said explicit dependence of the equation parameters on planarcoordinates is decoupled from the dependence on optical path difference;fitting said equation to global interference data acquired by varyingsaid optical path difference, thereby producing an optimal set of saidequation parameters; and using said optimal set of equation parametersto calculate a corrected phase.
 17. The method of claim 16, wherein saidfitting step includes establishing an error function based on saidequation and minimizing the error function with respect to said set ofequation parameters over the global interference data, thereby producingthe optimal set of equation parameters.
 18. The method of claim 17,wherein the step of minimizing the error function involves: (a)performing a first sub-stop of minimizing the error function withrespect to a first subset of said new phase-dependent parameters; (b)producing an updated error function with a first sub-set of thenew-phase-dependent parameters that minimized the error function duringsaid first sub-step; (c) performing a second sub-step of minimizing theupdated error function with respect to a second subset of said newphase-dependent parameters; (d) producing a further updated errorfunction with a second sub-set of the new phase-dependent parametersthat minimized the error function during said second sub-step; and (e)iteratively repeating steps (a) through (d) until a predeterminediterative condition is met.
 19. The method of claim 18, furtherincluding the step of calculating a height map of a sample based on saidcorrected phase.
 20. The method of claim 17, wherein said error functionis a least-squares representation of a difference between actualintensities produced by an interferometric system and theoreticalintensities predicted by said equation expressing intensity as afunction of planar coordinates, optical path difference, phase, and saidset of bandwidth-dependent and numerical-aperture-dependent equationparameters.
 21. The method of claim 16, further including the step ofcalculating a height map of a sample based on said corrected phase. 22.The method of claim 16, wherein said interference data acquired byvarying the optical path difference are taken symmetrically around abest focus position.
 23. The method of claim 16, wherein saidinterference data acquired by varying the optical path difference areproduced using a light source including a bandpass filter ofapproximately 20-nm bandwidth.
 24. The method of claim 16, wherein saidinterference data are acquired over three to twelve frames separated bya nominal phase step of π/2.
 25. The method of claim 16, wherein saidinterference data are acquired in a system producing about five fringeswithin a field of view of an objective of the system.
 26. The method ofclaim 16, wherein said step of varying the optical path difference iscarried out with a vertical scan.
 27. The method of claim 16, whereinsaid interference data acquired by varying the optical path differenceare taken symmetrically around a best focus position using a lightsource of approximately 20-nm bandwidth, and the interference data areacquired over three to twelve frames separated by a nominal phase stepof π/2.
 28. An interferometric method of profiling a sample surfacecomprising the following steps: performing an interferometricmeasurement of the surface, thereby producing a corresponding set ofinterferometric data; establishing an error function based on anequation expressing intensity as a function of planar coordinates,optical path difference, phase and a set of bandwidth-dependent equationparameters, said equation parameters having an explicit dependence onplanar coordinates, optical path difference and phase; replacing saidequation parameters with new phase-dependent parameters such that saidexplicit dependence of the equation parameters on planar coordinates isdecoupled from the dependence on optical path difference; minimizingsaid error function with respect to said equation parameters over theset of interferometric data, thereby producing an optimal set ofequation parameters; and using said optimal set of equation parametersto calculate a corrected phase.
 29. The method of claim 28, wherein saiderror function is a least-squares representation of a difference betweenactual intensities produced by said interferometric measurement andtheoretical intensities predicted by said equation expressing intensityas a function of planar coordinates, optical path difference, phase, andsaid set of bandwidth-dependent equation parameters.
 30. The method ofclaim 28, further including the step of calculating a height map of thesample based on said corrected phase.
 31. The method of claim 28,wherein: said error function is a least-squares representation of adifference between actual intensities produced by said interferometricmeasurement and theoretical intensities predicted by said equation; andsaid interferometric measurement is carried out by varying said opticalpath difference with a vertical scan with data-acquisition framesseparated by a nominal phase step of π/2.
 32. An interferometric methodof profiling a sample surface comprising the following steps: performingan interferometric measurement of the surface, thereby producing acorresponding set of interferometric data; establishing an errorfunction based on an equation expressing intensity as a function ofplanar coordinates, optical path difference, phase, and a set ofnumerical-aperture-dependent equation parameters, said equationparameters having an explicit dependence on planar coordinates, opticalpath difference and phase; replacing said equation parameters with newphase-dependent parameters such that said explicit dependence of theequation parameters on planar coordinates is decoupled from thedependence on optical path difference; minimizing said error functionwith respect to said equation parameters over said set ofinterferometric data, thereby producing an optimal set of equationparameters; and using said optimal set of equation parameters tocalculate a corrected phase.
 32. The method of claim 32, wherein saiderror function is a least-squares representation of a difference betweenactual intensities produced by said interferometric measurement andtheoretical intensities predicted by said equation expressing intensityas a function of and said numerical-aperture-dependent parameter. planarcoordinates, optical path difference, phase, and said set ofnumerical-aperture-dependent equation parameters.
 34. The method ofclaim 32, further including the stop of calculating a height map of thesample based on said corrected phase.
 35. The method of claim 32,wherein: said error function is a least-squares representation of adifference between actual intensities produced by said interferometricmeasurement and theoretical intensities predicted by said equation; andsaid interferometric measurement is carried out by varying said opticalpath difference with a vertical scan with data-acquisition framesseparated by a nominal phase step of π/2.
 36. A method of correctingeffects of modulation variations caused by illumination bandwidth andnumerical aperture in an interferometric system, the method comprisingthe following steps: developing an equation expressing intensity as afunction of planar coordinates, scanner position, phase, and a set ofbadwidth-dependent and numerical-aperture-dependent equation parameters,said equation parameters having an explicit dependence on planarcoordinates, scanner position and phase; replacing said equationparameters with new phase-dependent parameters such that said explicitdependence of the equation parameters on planar coordinates is decoupledfrom the dependence on scanner position; establishing an error functionas a least-squares representation of a difference between actualintensities produced by said interferometric system and theoreticalintensities predicted by said equation; minimizing said error functionwith respect to said set of equation parameters over global interferencedata acquired through an interferometric scan, said minimizing stepincluding: (a) performing a first sub-step of minimizing the errorfunction with respect to a first subset of said new phase-dependentparameters; (h) producing an updated error function with a first sub-setof the new-phase-dependent parameters that minimized the error functionduring said first sub-step; (c) performing a second sub-step ofminimizing the updated error function with respect to a second subset ofsaid new phase-dependent parameters; (d) producing a further updatederror function with a second sub-set of the new phase- dependentparameters that minimized the error function during said secondsub-step; and (e) iteratively repeating steps (a) through (d) until apredetermined iterative condition is met; thereby producing an optimalset of said equation parameters; using said optimal set of equationparameters to calculate a corrected phase; and calculating a height mapof a sample based on said corrected phase.